European University at St. Petersburg

THE METHOD OF QUANTILE REGRESSION,

A NEW APPROACH TO ACTUARIAL MATHEMATICS

Authors: Ruslan Abduramanov

Andrey Kudryavtsev

Saint Petersburg State University

INTRODUCTION

How correctly estimate a net premium?

Classical approaches:• An expected net premium.

• Risk loadings.

• Premium principles.

Disadvantages of classical approaches:• The problem of outliers.

• Fitting the theoretical distribution function.

• Independence of losses.

Advantages of the method of quantile regression:• In a series of observations, there is a small portion of

“outliers”.

• The distribution of the sample can be accurately fitted.

• Observations from the sample are not independent.

General model of quantile regression:

QUANTILE REGRESSION

'' :

'

:

' 11

miniiii xyi

iixyi

ii xyxyn

Quantile regression,

as a linear programming problem:

Several important consequences:• It is guaranteed that the computing procedure will be

finished with a finite number of iterations.

• Without changing estimates of s, can be increased/decreased to , robust.

QUANTILE REGRESSION

othercase

xyxyu

othercase

xyxyu

where

u

u

yuuX

uu

iiiii

iiiii

,0

,,0

,

,

0

0

min111

iy

/

-level of significance, then using well known quantile approach to rating:

where

In this case

So problem is to estimate a conditional distribution function

A NEW APPROACH TO ESTIMATENET PREMIUM RATES

,)()(:inf 1 XX FxxFxxP

)(]0Pr[]0Pr[)( xFXXxF YX

p

pFxP Y 1

1

)(xFY

The general model is as follow:

so, we actually build inverse conditional distribution function:

where А is all possible

As it is more convenient to take logarithmic function

Then receive:

A NEW APPROACH TO ESTIMATENET PREMIUM RATES

)(1 WβgY

),()( 11 Wβ gFY )1,0(

)(g )ln(

eeeeeY kkWW 11110 ...Wβ

The idea of new approach is to estimate quantile regression of type

with parameter

- Quantile is seen as a good tool to separate good and bad risks, thus, the method proposed gives adequate unbiased estimates.

- The avoidance of dividing the problem of the estimation of net premium rates into two subtasks – the assessment of expected net premium and the evaluation of risk loadings

A NEW APPROACH TO ESTIMATENET PREMIUM RATES

)()(1 WβgFY).(

Conditional premium rate calculated by the formula:

as a function , it is sufficient to take the expectation:

Then premium rate for the whole portfolio will be calculated with the following formula:

A NEW APPROACH TO ESTIMATENET PREMIUM RATES

))(( 1 Wβ gPY

)(

))(( 1Wβ gEPY

.)1()1(0 YY PpPppP

Analyzed data provided by one of the insurance companies operating on the motor insurance market in St. Petersburg (theft of vehicles): all contracts = 11790, where claims = 2359.

Let so

Characteristics of the object insured: the type of car, the colour, the region of theft and the date of theft.

-it is important to describe dependence of loss (claim size) from the characteristics of the object insured.

EXAMPLE

95.0 75.02.0

8.095.0

1

p

p

EXAMPLEHistogram (CrossSection.sta 14v*2359c)

1600,000014504,3478

27408,695740313,0000

53217,391366121,7391

79026,087091930,4348

1,0483E51,1774E5

1,3064E51,4355E5

Cena

0

100

200

300

400

500

600

700

800

900N

o o

f o

bs

Histogram of losses.

Assumptions about theoretical distribution function of losses:

1. Losses are LogNormally distributed.

2. Losses are Gamma-distributed.

-for both, the net premium estimates were achieved as a third quartile for appropriate constructed theoretical distribution.

By the method introduced in the paper, the model should be evaluated by quantile regression with parameter

-in the case of the method of quantile regression, it is enough to take expectation of the theoretical distribution.

EXAMPLE

75.0

EXAMPLE

The results of estimating by all methods, are shown in the table:

• The new method is free from some important assumptions which are critical for classical methods.

• The method proposed is based on quantiles which are popular for risk assessment in finance.

• The method proposed gives greater confidence to the actuary in the results received.

• The working capability and power of the method is shown with practical example on the real data.

CONCLUSION